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A masterpiece of Modern Logic: Kurt Gödel (1906-1978)


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A Masterpiece of Modern Logic:

Kurt Gödel (1906-1978)


Julie E. Goodman

MATH4010

March 17, 2005

The work of Kurt Gödel is often considered the most famous mathematical and philosophical piece of the twentieth century. His 1931 paper on the "Incompleteness Theorem" forever changed the meta-mathematical world in which every truth was thought to have been based upon a complete set of axioms. Kurt Gödel challenged the ideas of renowned mathematicians of his time and those before him to rethink and rearrange the rules of logic. The severe impact that Gödel’s conclusion had on mathematics was not realized immediately; it was too groundbreaking to be understood at first glance. However, unlike many celebrated genius’s, Gödel’s fame was recognized in his lifetime. From current day Czechoslovakia to New Jersey Kurt Gödel influenced the “who’s who” in science, mathematics, and philosophy and left a resolute impression on the academic world. The story of Kurt Gödel’s life is an intriguing tale of the complexities and hardships that go along with being one of history’s most impressionable intellectuals. To give a complete and concise account of our celebrated mathematician this paper will cover the most important details of Kurt Gödel’s life, from birth to death, with concentration on his academic contributions and the influence they have had on others.

To begin, it will be important to give a brief background of the Gödel family. Kurt’s parent’s, Rudolf Gödel and Marianne Handschuh, were brought up in the very same dwelling in Brünn, Moravia, current day Brno, Czechoslovakia. At the time it was part of the Austro-Hungarian Empire and a center for the textile industry, where Rudolf Gödel was employed. The population was predominantly Czech, however it had a large German-speaking minority to which the Gödel’s were included (Wang, 70). The German language proved to be a defining characteristic for the Gödel family, considering the political state of the country. The first of the two Gödel children was Rudolf, born in 1902 and a few years later on April 28, 1906 the “future mathematical genius” Kurt Gödel was born (Casti, 56).

Success in the textile industry allowed Rudolf Gödel to provide a very comfortable lifestyle for his family. When Kurt was seven the family moved into a custom-built home in the region of Spilberk (or Spielburg) where they had the luxury of a spacious garden and even a private guest-house (Casti, 56). Dawson describes the Spilberk area as a “fashionable neighborhood” due to its historical prestige as a 1740’s landmark for an old castle and fortress (7). Kurt and his brother were said to have played well together although they did not spend much time with other children (Dawson, 7). In an excerpt from a letter about their childhood, Rudolf noted that his brother was always a shy child when it came to being social and was very much attached to his mother (Casti, 56). Dawson writes, “Indeed, at age four and five, little Kurt reportedly cried inconsolably whenever his mother left the house”(6). Nevertheless, in school Kurt maintained a friendship with two boys by the names Harry Klepetar and Adolf Hochwald (Dawson, 14). One of the reasons Gödel had some social insecurities was because of his self-inflicted hypochondria, which began as early as 1911 but was further instigated by the rheumatic fever at the age of eight. Casti writes,

Although the disease seems to have left no lasting damage, the affliction was a turning point in Gödel’s life. His inquisitive nature caused him to read about the disease, and he thus learned about its possible side effects, including cardiac damage. Despite doctor’s assurances, he came to believe that his heart had indeed been affected, an unshakable conviction that he carried for the rest of his life (58).

Gödel was educated at a prestigious school from early on at the age of six. The first, Evangelische Privat-Volks- und Bürgerschule, was a private Protestant school that Gödel attended for four years. At ten years old Kurt and his brother attended the Realgymnasium where they both stayed until graduation. Realgymnasium was considered a public school however, it was still somewhat elitist because it charged tuition and had competitive entrance examinations. Most of the student body was German-speaking like the Gödel’s but there were few Protestant students as there were a considerable number of both Jews and Catholics (Dawson, 12). Of course Gödel excelled in all school subjects and rarely got an imperfect grade: “...for only once did he ever receive less than the highest mark in a subject,” ironically, that subject was mathematics (Dawson, 14)! Gödel placed a high importance on languages: at the Gymnasium he pursued Latin, French, and English and by the end of his lifetime Gödel’s repertoire of languages also included some Italian, Dutch, and Greek (Dawson, 15).

Kurt acquired the nickname “der Herr Warum,” which translates to “Mr. Why,” for his insistence on asking question after question (Casti, 56). Gödel’s acute interest in math can be traced back to about 1921, when he was 14 or 15, and studied an introductory calculus book in the “Göschen” collection (Wang, 72). In a letter to Hao Wang, Rudolf Gödel writes, “At sixteen to seventeen, he was far ahead of his classmates in mathematics and had already mastered the university material” (72). Realgymnasium was said to have had a renowned faculty but in a letter that Kurt wrote to his mother on September 11, 1960 he denounced the school entirely (Dawson, 18). According to Dawson,

...he attributed the awakening of this interest in mathematics and science not to his courses at school but to an excursion the family had made to Marienbad in 1921, when he was fourteen. There, he recalled, they had read and discussed Houston Stewart Chamberlain’s biography of Goethe, and it was Chamberlain’s account of Goethe’s color theory and his conflict with Newton, that in hindsight, Gödel felt had led directly to his own choice of profession (18).

After Realgymnasium in 1924, Kurt chose to join Rudolf and attend the University of Vienna where he began as a theoretical physics major. The decision to attend college in Vienna rather than Prague was significant to the Gödel’s affiliation with German-speaking cultures and their indifference to Slavic culture (Casti, 8). At the University of Vienna Gödel was mostly influenced by lectures given by Hans Han, Karl Menger, Philip Furtwängler, Moritz Schlick, Heinrich Gomperz and Rudolf Carnap (Casti, 60). Gödel’s decision to change his studies from theoretical physics to mathematics was made after about the first or second year at the university. The physics lectures Gödel attended were on the fourth floor of the Institute for Theoretical Physics and by coincidence the mathematics department was on the basement level of the same building... In 1926 Gödel chose to ‘move downstairs’ and concentrate his studies in mathematics (Casti, 59). A few sources agree that it was probably Furtwängler’s lectures on number theory that influenced Gödel in becoming a math major, and in turn can be attributed to the development of Gödel’s philosophy (Wang, 18; Dawson, 24).

By 1926 Kurt Gödel was invited to join Vienna’s academic elite in the “Vienna Circle” (Dawson, 26). It is said that he was most likely asked to join the group by Hans Hahn or Moritz Schlick, although many of Gödel’s professors attended the meetings. Upon being introduced to the “Vienna Circle,” whose meetings took place on Thursday nights in the Mathematics seminar room, Gödel was introduced to the work of Ludwig Wittgenstein (Dawson, 26). It is noteworthy that Gödel’s participation in the group was often stifled by his tendency to avoid controversy: he regularly found himself in disagreement with many of the ideas presented in the group but chose not to participate so as not to cause conflict (Casti, 61). In a 1975 questionnaire designed by Burke D. Grandjean Gödel replies to a question about his participation in the group: ‘Generally speaking I only agreed with some of their tenets. E.g. I never believed that math is syntax of lang. In the course of time I moved further and further away from their views’ (Wang, 20). When asked if the group had impacted his meta-mathematical philosophy, he states, ‘Yes, this group aroused my interest in the foundations but my views about them differ fund[amentally] (subst[antively]) from theirs’ (Wang, 17). Dawson writes, “But the Circle’s primary impact on him was in introducing him to new literature and in acquainting him with colleagues with whom he could discuss issues of common interest” (27). Gödel was only active in the group until about 1928 when he began to spend more of his time attending the colloquia of Karl Menger (Casti, 61). Kurt assisted Menger in editing a number of journals. In addition to editing, he contributed many of his own remarks and articles to the eight volumes (Dawson, 27).

While in college Gödel maintained a healthy social life, and his interests in the opposite sex blossomed. He was said to have always had a liking for older women, so it was no surprise that when he met his future wife, Adele Porkert, in 1927, she was six years older than him (Dawson, 34). Kurt’s parents were not particularly fond of Adele for she did not fit their criteria: she was Catholic, of a lower class, a divorcee, and above all, she was a dancer. However, Kurt’s father never saw the day of their marriage due to his untimely death on February 23, 1929 due to a prostatic abscess (Dawson, 33).

After their father’s death, Rudolf and Kurt chose to take their mother in and the three of them lived together in Vienna until the end of 1937. Casti writes, “Among Gödel’s many residences, this is the one best known by the scientific community. Between 1929 and 1937 he wrote his most important articles here, while carrying on correspondence with mathematicians from all over the world...” (63). It was there that Kurt wrote both parts of his dissertation on (in)completeness (to be discussed in detail in subsequent paragraphs).

Gödel received his doctorate of Philosophy in February of 1930 and by 1933 he was accepted as a “Privatdozent” (to further his degree), which allowed him to teach. His first course was on the foundations of arithmetic (Wang, xxi). At the end of 1933 he was honored by an invitation to lecture at the Institute for Advanced Study at Princeton in New Jersey. By February of 1934 he found himself traveling back and forth between America and Austria for many years. Unfortunately, his “Privatdozent” was retracted in 1939 at the request of a certain professor, most likely due to anti-Semitic sentiments (even though Gödel was not Jewish, he was associated with a minority crowd). The underlying political turmoil of WWII accounted for many changes in Gödel’s lifestyle both at the University and in Vienna in general. Nazi ideology affected Gödel’s circle of intellectuals, Jewish and non-Jewish, and in effect threatened Gödel himself:



Already in 1928, six years before the fascist takeover, most Austrian students identified with German nationalism, a tendency reflected in increasing efforts to break up seminars and lectures held by Jewish, socialist, and liberal, leftist professions. The attacks—particularly those against the Vienna Circle—became increasingly violent. They culminated in the murder of Moritz Schlick... Schlick was gunned down on June 22, 1936, on the steps of the University of Vienna (Casti, 78).

Between the tremendous effort in traveling from America to Austria, as well as the stress incurred by the political tension and evolving discrimination, Kurt’s mental health became more and more questionable. He spent a short period of time in a sanitarium in both 1931 and in 1934. In 1934, after being lonesome and depressed in Princeton, he had a nervous breakdown. His condition was treated by the renowned psychiatrist, Julius von Wagner-Jauregg (Moore, 899).

After Kurt and Adele were married in 1938, Kurt applied for another appointment of “Privatdozent” and received the position. He did not have much of an opportunity to take advantage of this position because he was called to military duty in 1939, despite a history of “precarious health” (Casti, 83). In order to avoid any further discrimination and the call to duty, Gödel and his wife moved once and for all to Princeton. They arrived in the U.S. on March 4, 1940 and never returned to their homeland.

The Gödel’s life in America was never the same. Even though Kurt gained his American citizenship (4/2/1948) he was always a foreigner. Gödel participated in the faculty life, but other than that he was very much an outsider. Nor did Adele make much of her social life, often blaming Kurt for her boredom (Casti, 91). The little social interaction that Kurt did participate in was almost exclusively with his close friends Albert Einstein and Oskar Morgenstern. Dawson notes that Kurt became so withdrawn over the years in America that he would purposely not leave his home when certain foreign visitors came to town (161). In Gödel: A Life of Logic, Kurt is described as having preferred phone conversations to human interactions, even when the conversation was with someone down the hall from him at the Institute (85). In 1966 he opted not to attend a symposium held by the Ohio Academy of Sciences in honor of his 60th birthday and his lifelong achievements.

Towards the end of his life Gödel’s mental health became increasingly bad. He materialized paranoia to the extent that it finally killed him. Afraid that someone would poison his food, he would only eat self-prepared meals. Not even Adele’s cooking would suffice. At one point Gödel’s weight was as low as 60lbs (Casti, 92). Kurt’s eating disorder coupled with an untreated abscess of his prostate resulted in severe illness (Moore, 901). When Adele finally brought Kurt to the hospital in on December 29, 1977, he died of starvation only two weeks later (Wang, xxvi). The death certificate attributed this to “‘malnutrition and inanition’ caused by a ‘personality disturbance’” (Moore, 901). Adele survived Kurt’s death by three years and in that time she was able to collect the many boxes of Kurt’s books and papers, known as the scientific Nachla, and donated them to the Institute for Advanced Study in his memory (Dawson, 256).

To begin the description of Gödel’s significant contributions to the mathematical world, it should first be noted that Gödel considered himself a “Platonist.” In relation to mathematics, this philosophical view “has for its subject-matter a realm of real, non-spatial, non-mental, timeless objects” (Barker, 2). Gödel considered all concepts to have an objective reality of their own, that cannot be created or changed, but can only be perceived and described (Dawson, 199). This is most important in understanding the developments made by Gödel that are discussed in the following paragraphs. Over Gödel’s lifetime he wrote a number of scholarly papers, gave influential lectures all over the world, consulted with the most brilliant of his contemporaries, and was honored in many ways for his numerous achievements. The most significant of these shall be discussed here: the incompleteness theorem and his work on the axioms of choice and the continuum problem.

Gödel’s research on the completeness of formalized logic is what he is most celebrated for. He developed upon Bertrand Russell and Alfred North Whitehead’s Principia Mathematica (1910) to establish the conclusion that the ancient idea of the “axiomatic method” contains in it inherent limitations when used in formalized systems. His interest in this subject stemmed from a problem dealt with in Hilbert and Ackermann’s Grundzüge der theoretische Logik, that all assertions should be provable in a formal system (Casti 48, Moore, 901).

Gödel’s “Incompleteness Theorem” expressed the limits to the power of logical deduction. The most succinct statement of this theorem is that any system powerful enough to form statements about itself is either inconsistent or incomplete. Basically, “Gödel discovered that it is possible for a statement in a formal system not only to talk about itself, but also to deny its own theoremhood” (Hofstadter, 3). The simplest way of describing such an example of self-referentialism is the well-known barbershop example. In this example there is a certain barber in a town, who for every person in the town that does not shave himself is shaved by the barber. The confusion arises when we consider who shaves the barber? If the barber does not shave himself then the barber must shave himself; a contradiction.



This theorem obviously has consequences beyond the simple example of the barber. In order for Gödel to explain that this limitation exists for all formal systems, he created a system based on the Principia Mathematica where the “various notions employed in mathematical analysis are definable exclusively in number-theoretical terms” (Nagel, 42). He assigned unique “Gödel Numbers” to each elementary sign that would represent every character used in constructing a mathematical sentence. The following chart is an example of the “Gödel Numbers” as shown in Nagel and Newman’s Gödel’s Proof:

Constant Sign

Gödel Number

Usual Meaning

~

1

Not



2

or



3

If....then...



4

There is an...

=

5

Equals

0

6

Zero

s

7

The immediate successor of

(

8

Punctuation mark

)

9

Punctuation mark

,

10

Punctuation mark

+

11

Plus



12

Times

With these new definitions for characters Gödel constructed meaningful sentences just out of the numbers that represent the characters. The actual numbers could be represented by another numerical system where the integers correspond to the prime numbers greater than 11. One very simple example would be: (1+1=2) is equivalent to 8,13,11,13,5,17,9. The true “Gödel Numbers” are more complex than this and when statements of mathematical truths are converted into the “Gödel Numbers,” they create a complete number of their own. Gödel showed that in this type of system there is an infinite class of theorems and that by using the table of meanings (as the one above) there will be an infinite class of numerical truths that can be converted into a formal statement and hence a theorem of the new system. In this way, Gödel showed that every expression in the created system could be shown by a meta-mathematical statement in a corresponding “Gödel Number.” And thus meta-mathematics could become completely ‘arithmetized’ (Nagel, 80).

In essence, Gödel then created a system that was at first glance considered complete. However, in a complete system there cannot be contradiction. Moreover, the system cannot produce a mathematical truth that is provable and also show that its negation is provable. What Gödel showed was that the hypothetical systems would always produce a statement about itself that could be both proved and disproved, and was hence inconsistent and in turn, incomplete. This is the self-recursive issue that finally disproved Hilbert and Ackermann’s idea that formal mathematics could eliminate logical paradoxes. At the Ohio Academy of Science’s symposium honoring Kurt Gödel, Steven Barker said,



From the viewpoint of a realistic philosophy of mathematics, the incompletability theorem can be regarded not as calling into question the independent reality of mathematical entities such as sets or numbers, but rather as indicating an essential limitation in the expressive power of symbolism; the limitation being that no symbolism can fully succeed in characterizing a system of objects as rich as the natural numbers (4).

It should be noted that when Gödel introduced the first part of the incompleteness idea to the public at a conference in Könisberg on October 5-7, 1930 it’s greatness was not immediately recognized. Hintikka writes, “Gödel’s result was so new and so puzzling methodologically that it did not sink in immediately. The speaker who tried to sum up the discussion did not even mention Gödel’s result” (4). The only exception was John von Neumann who understood immediately the implication of such an announcement.

The Axiom of Choice and the Continuum Hypothesis were Gödel’s main focus while at Princeton, despite the fact that he had cut back his direct work on mathematical logic at this point in his life (Casti, 85)1. Gödel challenged the 1904 work of Ernst Zermelo that formulated the axiom of choice as an explicit assumption. Although Gödel could not prove that these axioms were independent of the other axioms of set theory, his contributions to the subject are still considered a masterpiece of classic modern mathematics. When this was finally proved in 1963 by Paul J. Cohen (published 1964) it was largely due to the work Gödel had done before him.

1 The axiom of choice states that whenever each member of a set is a non-empty set, and no pair of numbers in the set share a common element, then there exists another set that contains exactly one element from each member of the first set, this being the “choice set.” The continuum hypothesis is a statement that asserts that the cardinality of the continuum is the smallest uncountable cardinal number. This hypothesis states that there does not exist a set of size intermediate between the natural numbers and the continuum) (“Axiom of Choice”).

In conclusion, the work of Kurt Gödel has shaped the world of meta-mathematics in definitive ways. “Mr. Why” shook the mathematical world with his statement and proof of a remarkable theorem that set down the limitations of mathematics, the “Incompleteness Theorem.” He examined the possibilities of completeness and consistency in math logic until he was able to make a conclusion that could be proved; leaving no more room for debate. The discovery of such limitations in mathematics has been revolutionary to math logic, and Kurt Gödel will forever be remembered for this and his many other contributions.


Bibliography

“Axiom of Choice.”

Barker, Stephen F. “Realism as a Philosophy of Mathematics.” Bulloff 1-9.

Bulloff, Jack J. ed. Foundations of Mathematics: Symposium Papers Commemorating the Sixtieth Birthday of Kurt Gödel. New York: Springer-Verlag, 1969.

Casti, John L., and Werner DePauli. Gödel: A Life of Logic. Massachusetts: Persus Publishing, 2000.

Dawson, John W. Logical Dilemmas: The Life and Work of Kurt Gödel. Massachusetts: A K Peters, 1997.

Hintikka, Jaakko. On Gödel. California: Wadsworth, 2000.

Hofstadter, Douglas. “Kurt Gödel.” Time Magazine. Mar. 29 1999. .

Katz, Victor J. A History of Mathematics: An Introduction. 2nd ed. Massachusetts: Addison-Wesley Educational Publishers, 1998.

Moore, Gregory H. “Gödel, Kurt Friedrich.” Biographical Dictionary of Mathematicians. Vol. 2. New York: Charles Scribner’s Sons, 1991.

Nagel, Ernest, and James R. Newman. Gödel’s Proof. New York: New York University Press, 2001.

Wang, Hao. Reflections on Kurt Gödel. Massachusetts: The MIT Press, 1987.


“Axiom of Choice.”


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